Numbers - 01

Explanation:-

Unit digit in 3⁴ = 1 ⇒ Unit digit in (3⁴)¹⁶ = 1
Therefore Unit digit in 3⁶⁵ = Unit digit in [ (3⁴)¹⁶ x 3 ] = (1 x 3) = 3
Unit digit in 6⁵⁹ = 6
Unit digit in 7⁴ ⇒ Unit digit in (7⁴)¹⁷ is 1.
Unit digit in 7⁷¹ = Unit digit in [(7⁴)¹⁷ x 7³] = (1 x 3) = 3
Therefore Required digit = Unit digit in (3 x 6 x 3) = 4.

Explanation:-

$$\eqalign{ & 20 \times x = \left( {64 + 36} \right)\left( {64 - 36} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 100 \times 28 \cr & \,\,\, \Rightarrow \,x\, = \frac{{100 \times 28}}{{20}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 140 \cr}$$

Explanation:-

Let S_n = (2 + 4 + 6 + ..... + 30). This is an A.P. in which a = 2, d = 2 and l = 30
Let the number of terms be n Then,
a + (n - 1)d = 30
⇒ 2 + (n - 1) x 2 = 30
⇒ n = 15
$$\therefore {S_n} = \frac{n}{2}\left( {a + I} \right) = \frac{{15}}{2} \times \left( {2 + 30} \right)$$
= (15 x 16) = 240

Explanation:-

437 > 22
All prime numbers less than 22 are : 2, 3, 5, 7, 11, 13, 17, 19.
161 is divisible by 7, and 221 is divisible by 13.
373 is not divisible by any of the above prime numbers.
Therefore373 is prime.

Explanation:-

Given Exp.
= (397)² + (104)² + 2 x 397 x 104
= (397 + 104)²
= (501)² = (500 + 1)²
= (500²) + (1)² + (2 x 500 x 1)
= 250000 + 1 + 1000
= 251001

Explanation:-

$$\eqalign{ & x = 13p + 11\,{\text{and}}\,x = 17q + 9 \cr & \therefore 13p + 11 = 17q + 9 \cr & \Rightarrow 17q - 13p = 2 \cr & \Rightarrow q = \frac{{2 + 13p}}{{17}} \cr & {\text{Thr}}\,{\text{least}}\,{\text{value}}\,{\text{of}}\,p\,{\text{for}}\,{\text{which}} \cr & q = \frac{{2 + 13p}}{{17}}\,{\text{is}}\,{\text{a}}\,{\text{whole}}\,{\text{number}}\,{\text{is}}\,p = 26 \cr & \therefore x = \left( {13 \times 26 + 11} \right) \cr & \,\,\,\,\,\,\,\,\,\,\, = \left( {338 + 11} \right) \cr & \,\,\,\,\,\,\,\,\,\,\, = 349 \cr}$$

Explanation:-

Let the given number be 476 xy 0
Then (4 + 7 + 6 + x + y + 0) = (17 + x + y) must be divisible by 3
And, (0 + x + 7) - (y + 6 + 4) = (x - y -3) must be either 0 or 11
x - y - 3 = 0     ⇒     y = x - 3
(17 + x + y) = (17 + x + x - 3) = (2x + 14)
⇒ x = 2 or x = 8
⇒ x = 8 and y = 5

Explanation:-

Let x be the number and y be the quotient. Then,
x = 357 × y + 39
   = (17 × 21 × y) + (17 × 2) + 5
   = 17 × (21y + 2) + 5
∴ Required remainder = 5

Explanation:-

24 = 3 x 8, where 3 and 8 co-prime
Clearly, 35718 is not divisible by 8, as 718 is not divisible by 8
Similarly, 63810 is not divisible by 8 and 537804 is not divisible by 8
Consider option (D),
Sum of digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3
Also, 736 is divisible by 8
∴ 3125736 is divisible by (3 x 8), i.e., 24

Explanation:-

72519 x 9999
= 72519 x (10000 - 1)
= 72519 x 10000 - 72519 x 1
= 725190000 - 72519
= 725117481